Particle Impact Analysis of Bulk Powder During
Pneumatic Conveyance
Report Contributors: Giles Richardson1 , Mark Cooker2 ,
Niamh Delaney3 , Kevin Hanley4 , Poul Hjorth5 , Dana Mackey6,
Joanna Mason3,7, Sarah Mitchell3 and Catherine O’Sullivan8
Industry Representative: Seamus O’Mahony9
1
School of Mathematics, University of Southampton, United Kingdom
School of Mathematics, University of East Anglia, United Kingdom
3
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
4
Department of Process and Chemical Engineering, University College Cork, Ireland
5
Department of Mathematics, Technical University of Denmark, Denmark
6
School of Mathematical Sciences, Dublin Institute of Technology, Ireland
7
Report coordinator, [email protected]
8
Department of Civil and Environmental Engineering, Imperial College London, United Kingdom
9
Wyeth Nutritionals Ireland, Askeaton, Co. Limerick, Ireland
2
Abstract
Fragmentation of powders during transportation is a common problem for manufacturers of
food and pharmaceutical products. We illustrate that the primary cause of breakage is due to
inter-particle collisions, rather than particle-wall impacts, and provide a statistical mechanics
model giving the number of collisions resulting in fragmentation.
86
1
Introduction
Pneumatic conveyance of food and pharmaceutical products in dehydrated, powdered form, is
very common in Ireland’s process industries. During conveyance, powder particles may break
into smaller pieces, due to impacts, resulting in a change in overall bulk density. A leading
manufacturer of infant formula approached the 70th European Study Group with Industry to
investigate the change in bulk density that they observe during transportation of this powder.
The powder is inhomogeneous and consists of approximately six main ingredients, and around
twenty additional micro-nutrients, which are smaller particles. An average particle is approximately spherical and around 100 microns in diameter. Particles consist of proteins, sugars and
fats and can be either solid or hollow. If hollow, the size of the wall is typically one fifth of the
diameter, and the walls can be solid or porous see Figure 1a.
Initially the powder is produced from a liquid in the spray dryer. During this phase the particles
can coalesce into clusters (see Figure 1b), which is desirable as agglomerates of particles are
better at dissolving in water, when reconstituting the infant formula. The particles are blown
along pipes by an air flow, and then intermediately stored in silos, before packing into cans.
During transportation particles impact both the walls of the pipe and also each other. A
schematic diagram of an industrial pneumatic conveyance line is shown in Figure 2. Typical
parameters for both the particles, and the pneumatic conveyance line are listed in Table 3.
Infant formula is usually measured out by volume using a scoop. Therefore bulk density of
the powder is an essential quality control parameter for infant formula manufacturers. If there
is too much breakage during transportation, the average size of the particle clusters decreases
because the particles can pack together more tightly, and as a result the bulk density increases.
This results in the formula becoming too nutrient rich when reconstituted. Conversely, if there
is unusually little breakage during transportation, the final bulk density could be too low, and
the rehydrated formula will be insufficient in meeting the infant’s nutritional requirements.
As a certain amount of breakage during transportation is inevitable, manufacturers currently
estimate the change in bulk density from empirical data. The aim of the study group is to devise
a mathematical model for the amount of breakage during transportation, and thus quantify the
bulk density change.
The purpose of this report is twofold: to outline the main mechanisms of impact, and to provide
an analytical expression for the rate of particle breakage during transportation. The outline
is as follows. We begin in Section 2 by describing Dean flow, which is the standard way of
describing flow in a curved pipe. In Sections 3 and 4 we consider particle flow in the pipe.
We initially consider a toy two-dimensional problem, and then extend our analysis to the full
three-dimensional problem. An overview of Hertzian contact laws, which we can use to estimate
the energy at which particle impacts result in fracture, is presented in Section 5. For Hertzian
theory to apply, it is assumed that the impacting bodies are solid. In Section 5.2 we outline an
argument to illustrate that the walls of the particles under consideration are sufficiently thick
that we may justifiably model them as solid. Section 6 focuses on constructing a statistical
87
M
Figure 1: SEM images of a typical particle and clusters of particles. A scale bar is shown below
each image. Photographs reproduced with permission from Kevin Hanley.
3m
Tall−form
Intermediate
spray
storage silo
dryer
Air out
20m
0.1m
Fluidised bed
5m
Air in
Rotary feeder
To
Air input
Pneumatic conveying line
canning
line
Figure 2: A schematic diagram of an industrial pneumatic conveyance line for the transport of
infant formula from the spray drying phase to the storage silos.
88
Quantity
Particle radius
Radius of pipe
Radius of curvature of pipe
Estimated density of particles in the pipe
Estimated velocity of particles in the pipe
Poisson’s ratio
Young’s Modulus
Coefficient of restitution
Yield stress
symbol
a0
R
κ0
ρ
v0
ν
E
σf
value(s)
5 × 10−6 –5 × 10−4
0.05
1
250–350
0.5–1
0.3
3 × 109 (from [10])
0.2
5 × 105
units
m
m
m
kg/m3
m/s
Pa
Pa
Table 3: Typical particle and pipe parameters provided by Kevin Hanley, unless otherwise
stated.
mechanics model of collisions resulting in fragmentation, which implicitly incorporates Hertzian
theory. Finally, in Section 7 we provide some concluding remarks and outline areas for future
work.
2
The flow in the pipe
The pipe has an approximately circular cross-section and has a centreline that is weakly curved
(i.e. has typical radius of curvature L ≫ 2σ where 2σ is the diameter of the pipe cross section.
Flow in curved pipes is usually described as a Dean flow [4] and reviewed in [3]. Such flows are
characterised by the so-called Dean number De and δ the ratio of the typical pipe radius (or halfwidth in the case of non-circular pipes) σ to its typical radius of curvature L. Unfortunately
the Dean number is not defined in the same manner in all works. Here we shall adopt the
definitions
ρGσ3 (2σ/L)1/2
De =
,
µ2
δ=
σ
,
L
(2.1)
where G is a typical pressure gradient along the pipe, µ is the viscosity of the fluid and ρ
its density. At low Dean numbers (below about 100) this definition is equivalent to another
commonly used definition namely
1/2
2σρW̄0
^ = 4Re 2σ
De
,
(2.2)
where the Reynolds number Re =
L
µ
where W̄0 is the mean velocity along the pipe. However these two definitions do not coincide
for larger Dean numbers.
89
At low Dean numbers De ≪ 1, and for loosely coiled pipes δ ≪ 1, the flow is, to a good
approximation, a Poiseuille flow along the axis of the pipe with a small circulatory component
in the transverse cross-section of the pipe. In the study group we initially made the assumption
that significant flow (as far as motion of particles in the flow is concerned) occurs mainly in the
axial direction. However, as we shall briefly outline here, this assumption is probably invalid
so that further investigation should be directed to considering flow in the cross-section of the
pipe. In fact the high Dean number limit turns out to be of more relevance to the flows in the
conveyance line than the low Dean number limit.
The structure of the flow at high Dean number. The high Dean number limit has been
considered previously by [5, 6, 14] by using a mixture of numerical and asymptotic techniques.
They define the flow velocity u = wt + v(sin αb − cos αn) + u(cos αb + sin αn) (here t is the
unit vector along the axis of the pipe, n is the unit vector in the plane of the pipe centreline
directed towards the inside of the this curve, b is the other unit normal vector (see Figure 3)).
In the limit De ≫ 1 δ ≫ 1 [5, 6, 14] have shown that in the bulk of the flow the scalings for
the various velocity components are
1/2 !
ν
ν
ν
L
u = O De1/3
, v = O De1/3
, w = O De2/3
.
(2.3)
σ
σ
σ 2σ
In addition they identify a boundary layer of thickness O(σDe−1/3 ) around the edge of the pipe
in which the solution is such that
1/2 !
ν
ν
ν
L
u = O De2/3
, v = O De1/3
, w = O De2/3
.
(2.4)
σ
σ
σ 2σ
At these high Dean numbers the Dean number is related to the mean axial velocity W̄ by (2.3c),
that is by
3/2 9/4 W̄ σ
.
(2.5)
De = O
ν3/2 L3/4
Here we give no further details of this solution except to say that it has to be calculated
numerically, even in the boundary layer.
Turbulent pipe flow. The conveyance line flow occurs at high Reynolds number. So for
example, we estimated the Reynolds number as 20,000 by taking the viscosity of the flow to be
that of air. In practice the net viscosity of the mixture of air and particles will be more than
this, so the Reynolds number will be less than for air alone. Nevertheless the flow may still be
turbulent. The boundary layer structure of a turbulent flow in a straight pipe is described to a
reasonable approximation by the so-called Law of the Wall (see for example [1, 16]). It is less
clear however what the equivalent structure is in a curved pipe.
90
M
b
u
v
α
n
(a)
t
(b)
Figure 3: Illustration of the coordinates used to describe the flow in the pipe. Panel (a)
shows the definitions of the vectors {t, n, b} while panel (b) shows the definition of the velocity
components (here t is directed into the paper).
3
Can collisions of particles with the wall of the pipe account for particle fragmentation?
Here we consider particle motion in the pipe flow. Since particles are small such motion is
usually dominated by the fluid drag, rather than inertia. This motivates us to ask the question
can particles ever collide with the side of the pipe with sufficient velocity in order to cause their
fragmentation (i.e. velocities of O(1ms−1 ))? In particular we note that it is the centrifugal
force on the particle that is likely to be the cause of motion across streamlines towards the edge
of the pipe and, further, that since the flow velocity at the boundary of the pipe is zero we
would not expect the centrifugal force on a particle close to the edge of the pipe to be large.
However it is still possible to envisage a situation in which there is a narrow boundary layer
in the vicinity of the pipe wall, across which the fluid velocity drops rapidly to zero, where
the particle has sufficient inertia to cross from the region of high fluid velocity to the pipe wall
without losing most of its kinetic energy. In order to investigate this possibility we will consider
the toy problem of a particle moving in a two-dimensional flow.
3.1
The 2D problem
Here we consider a toy two-dimensional problem noting that where the two-dimensional boundary layer structure is similar to that of the real flow the resulting particle trajectories will be
qualitatively similar. Since the particle Reynolds number is typically small we can approximate
91
its equation of motion by balancing particle inertia against linear Stokes drag to give
d2 x
dx
m 2 = 6πµrp v −
.
dt
dt
(3.1)
Here m is the mass of one particle, rp is the radius of the particle, x and v are the position
and velocity of the particle respectively, and µ is the fluid viscosity. Note that the Stokes drag
formula is assumed because the flow of air relative to the moving particle has a low (local)
Reynolds number, quite different from the high Re for the air flow down the pipe.
We would like to consider two particular scenarios. In the first (i) the particle moves in a
pipe flow around a circular bend in which the predominant flow is in the direction of the
tangent to the centreline t (i.e. takes the form u = wt). The second scenario (ii) aims to
model the situation where the secondary Dean flows in the cross-section of the pipe are of more
significance to particle-wall collisions than the primary flow along the axis of the pipe. In order
to model these situations we introduce polar coordinates for the particle describing its position
by x = R(t)er so that, on differentiation with respect to t we find
ẋ = Ṙer + Rθ̇eθ ,
ẍ = (R̈ − Rθ̇2 )er + (2Ṙθ̇ + Rθ̈)eθ ,
where er and eθ are the unit radial and azimuthal vectors in the coordinate directions of the
plane polars r and θ, whose origin is at the pipe centreline. Furthermore in both scenarios
(i) and (ii) the flow is predominantly in the azimuthal direction; this motivates us to write
v = g(R, θ)eθ .
We non-dimensionalise (3.1) by writing
x = R0 x∗ , R = R0 R∗ , v = V0 v∗ , t =
R0 ∗
t,
V0
which gives, on dropping ′ ∗s, substituting x∗ = R∗ er and equating components in the radial
and azimuthal directions,
R̈ − Rθ̇2 = −Γ̃ Ṙ,
2Ṙθ̇ + Rθ̈ = Γ̃ (g(R, θ) − Rθ̇).
(3.2)
where
Γ̃ =
6πµrpR0
.
mV0
In scenario (i) we are typically interested in a pipe with radius of curvature which is much
greater than the pipe radius whereas in scenario (ii) we are interested in flow occurring in a
narrow boundary layer around the edge of the pipe. This suggests that we should further write
R = 1 − εz where R = 1 denotes the outer edge of the pipe and where the small parameter
ε = XRbl0 gives the ratio of the boundary layer thickness to the typical radius of curvature of the
92
pipe. On making this substitution in (3.2), writing θ̇ = Ω and on rescaling time by writing
t = εΓ̃ τ we obtain the following system
d2 z
dz
= −(1 − εz)Ω2 − Υ 2
dτ dτ
dΩ
dz
Υ (1 − εz)
=g
^(z, θ) − (1 − εz)Ω
− 2εΩ
dτ
dτ
dθ
= ε1/2 Υ−1/2 Ω.
dτ
(3.3)
(3.4)
(3.5)
where Υ = ε−1 Γ̃ −2 . The key parameter in this problem is Υ (recall that ε ≪ 1). This parameter
corresponds to the ratio of to the timescale for drag to accelerate a particle to the velocity of the
flow to that for a particle to traverse the boundary layer. Thus if Υ ≪ 1 the particle moves, to
a good approximation, with the velocity of the flow. If however Υ = O(1) the particle can cross
the boundary layer to the wall of the pipe without being decelerated to almost zero velocity at
the wall. In other words if Υ ≪ 1 it will impact the wall (at z = 0) with very small velocity
(an asymptotic analysis reveals that the dimensionless velocity normal to the wall on impact
will at most be O(Υ2 ) corresponding to a dimensional velocity of at most O(V0 ε1/2 Υ5/2 )). For
Υ = O(1), however, the particle impacts the wall with O(1) dimensionless velocity normal to
the wall corresponding to a dimensional velocity of O(V0 ε1/2 ). 10
In summary we expect that the chance of particle impacts on the wall leading to particle
fracture is negligible if
Υ=
m2 V02
≪ 1,
(6πµrp)2 R0 Xbl
where Xbl is the width of the flow boundary layer and R0 is the typical radius curvature of
the problem (in case (i) flow along the axis of the pipe R0 is the typical radius of curvature of
the pipe centreline while in case (ii) in which secondary flows dominate the process R0 is the
radius of the pipe) . Where Υ = O(1) the particle impacts the wall with a velocity of O(V0 ε1/2 )
and there is a significantly greater chance of particle fracture on impact. Even where Υ ≪ 1
inter-particle collisions give another possible mechanism giving rise to particle fracture. This
will be considered further in §4.
Relevance to Dean flow. Consider first the flow along the pipe, parallel to its centreline,
corresponding to case (i). Here we can identify R0 with the radius of curvature of the pipe
centreline L, the velocity can be read off from (2.3c) and (2.4c) so that the key parameters are
1/2
L
2/3 ν
R0 = L,
V0 = De
,
Xbl = σDe−1/3 ,
σ 2σ
10
Note that where Υ ≫ 1 the asymptotic analysis reveals an inertial particle trajectory in which both w and
dz/dτ are unchanged to leading order; in order to retain more information it is necessary to rescale distance
normal to the boundary to give an O(1) value of Υ.
93
u
u
R
θ
R
θ
(i)
(ii)
Figure 4: The flows considered for the toy problem
and thus give
Υ=
m2 ν2 De5/3
(6πµrp)2 2σ4
in case (i).
(3.6)
In case (ii) we can identify R0 with the radius of the pipe σ, while the typical velocity is that
in the azimuthal direction (which can be read off from (2.4a)). The key parameters are thus
R0 = σ,
ν
V0 = De2/3 ,
σ
Xbl = σDe−1/3 ,
from which it follows that
m2 ν2 De5/3
Υ=
(6πµrp)2 σ4
in case (ii).
(3.7)
We can relate these expressions for Υ to the mean axial velocity in the pipe W̄ via (2.5). It
is immediately apparent that Υ is the same order of magnitude in both cases signifying that
the azimuthal and longitudinal velocities occurring in Dean flow have comparable effects on
the motion of particles across the boundary layer adjacent to the pipe boundary. Thus any
quantitative analysis requires that we consider the full three-dimensional problem (rather than
the toy model treated here).
94
4
Particle transport in the pipe
The particle Reynolds number is relatively small and so we can approximate drag force by the
linear Stokes relation. Balancing inertia with the drag force gives
d2 x
dx
m 2 = 6πµa v −
,
(4.1)
dt
dt
where a is the radius of the particle, x and v are the position and velocity of the particle
respectively, and µ is the fluid viscosity.
Since the radius of the pipe is generally considerably smaller than its radius of curvature it is
helpful to introduce a local coordinate system based about the centre line of the pipe, r = q(s),
defined by
(4.2)
x = q(s) + ξn(s) + ηb,
where n(s) and b are the principal unit normal
p and unit binormal vectors respectively, s
measures arclength along the centre line, and η2 + ξ2 measures the distance of the point
from the centre line.
We use the standard Serret-Frenet formulae (see, for example [12]) to relate the various quantities in (4.2), namely
dq
= t,
ds
dt
= κn,
ds
dn
= τb − κt,
ds
(4.3)
where κ is the pipe curvature, τ is the torsion and t is the tangent vector. To make the equations
complete we also have db/dt = τn. We assume a planar pipe and so τ = 0, therefore in the
subsequent analysis we can ignore the n component of the equations of motion.
In order to track an individual particle at position x(t) we rewrite (4.2) as x(t) = q(s(t)) +
ξ(t)n(s(t)) + ηb and use (4.1) to find differential equations for ξ and s. Before substituting
x(t) into (4.1) we nondimensionalise by choosing
x=
x∗
,
κ0
q=
q∗
,
κ0
r = Rr∗ ,
ξ = Rξ∗ , a = a0 a∗ , t =
η = Rη∗ , κ = κ0 κ∗ ,
t∗
, v = v0 v∗ ,
κ0 v0
s=
s∗
,
κ0
(4.4)
where R is the radius of the pipe, κ0 is the typical curvature of the pipe centreline, a0 is a
typical particle radius and v0 is a typical fluid velocity. On dropping the ∗’s the dimensionless
equations (4.1) and (4.2) become
d2 x
dx
Γ
,
(4.5)
= 2 g(r)t −
dt2
a
dt
95
and
x = q(s) + ǫ(ξn(s) + ηb),
(4.6)
where we have assumed that the flow is primarily
p parallel to the centreline of the pipe and has
the nondimensional form v = g(r)t, with r = η2 + ξ2 . We assume that the coordinate η is
constant while the particle moves. Note that equation (4.3) remains unchanged.
The dimensionless parameters in (4.5) and (4.6) are given by
Γ=
9µ
2a20 v0 κ0 ρ
ǫ = Rκ0 .
,
(4.7)
Typical sizes of these parameters, using Table 3, are Γ ≈ 11 and ǫ ≈ 0.1, which allows us to
set Γ = γ/ǫ where γ = O(1).
Using the Serret-Frenet equations (4.3) we differentiate x in (4.6) to give
dx
= ṡ(1 − ǫξκ)t + ǫξ̇n
dt d2 x
dκ
2
s̈(1
−
ǫξκ)
−
ǫ
ṡ
2
ξ̇κ
+
ξ
=
ṡ
t
+
κ(1
−
ǫξκ)
ṡ
+
ǫ
ξ̈
n.
dt2
ds
(4.8)
(4.9)
Upon substitution into (4.5) and equating coefficients of t and n we obtain the following
equations relating ξ and s
γ
dκ
[g(r) − ṡ(1 − ǫξκ)]
(4.10)
s̈(1 − ǫξκ) − ǫṡ 2ξ̇κ + ξ ṡ =
ds
ǫa2
γ
κ(1 − ǫξκ)ṡ2 + ǫξ̈ = − 2 ξ̇.
(4.11)
a
Assuming nǫ ≪ 1 we find that the leading order terms satisfy
p
a2 κ(s) 2
ṡ .
(4.12)
ṡ = g( η2 + ξ2 ),
ξ̇ = −
γ
Eliminating ṡ gives the expression
i2
a2 κ(s) h p 2
g( η + ξ2 ) ,
γ
(4.13)
i2
2a2 κ(s)v20 ρ h p
g( (η/R)2 + (ξ/R)2) .
9µ
(4.14)
ξ̇ = −
or in dimensional form
ξ̇ = −
Further work could treat a particle on the median plane η = 0. Also quite generally, since
decreases to zero as the particle
g(r = 1) = 0 at the wall, we can at least be sure that dξ
dt
2
approaches the wall. Although we can’t set g(r) = 1 − r , because the Reynolds number is
high, such a dependence in the boundary layer near r = 1 tells us that 1 − r = 1 − ξ(t) (on
η = 0) decreases to zero with exponential decay, hence there is no impact velocity.
96
5
5.1
Hertz theory for binary collisions
Critical Energy
Hertz theory is an analysis of stresses at the contact of elastic solids. It was initially developed
for static loading, but subsequently extended to quasi-static impacts (e.g. of spheres). We use
Hertz theory to give an estimate for the critical energy, i.e., the energy needed to break a given
particle of some radius a, here we give a brief overview of the key equations.
As in [15], consider a (collinear) collision between two particles, modelled as smooth, homogeneous, solid, elastic spheres, of radii a1 and a2 respectively. The elastic properties of the
spheres are given by their Young moduli Ei and Poisson’s ratios νi (i = 1, 2). We look at a
quasi-static process and thus consider an instant where the mutual force of compression is P.
The spheres compress elastically, with a contact area of radius ac , see Figure 5.
P
z
Figure 5: From left to right: elastic sphere forced onto an elastic plane with a force P, and two
spherical particles of radii a1 and a2 colliding collinearly. z is the amount of compression, and
ac is the radius of the contact disk.
The distribution of pressure across the contact area, as a function of distance r away from the
centreline, is
r
p = p(r) = p0 1 −
,
(5.1)
ac
where ac is the radius of the contact zone between the particles, and the maximum pressure p0
is given by
3P
p0 =
.
(5.2)
2πa2c
97
The effective values of the Young’s modulus E∗ and radius a∗ are defined via
1 − ν1 2 1 − ν2 2
1
+
,
=
E∗
E1
E2
1
1
1
+
,
=
a∗
a1 a2
and
respectively.
From Hertz theory, we then have the following relations between the applied force P, the
effective Young’s modulus E∗ , the effective radius a∗ , and the compression distance z:
4
P = E∗ a∗1/2 z3/2 ,
3
where z is given by a2c = a∗ z, and
ac =
3Pa∗
4E∗
13
.
(5.3)
(5.4)
The critical compression distance, at which fracture occurs, is denoted zf . For a collision that
just reaches this critical compression length zf , the work done by the compression force just
equals the initial kinetic energy Er , and thus:
Z zf
1
2
Er = mVf =
P(z)dz.
(5.5)
2
0
V is the relative impact velocity, and Vy is defined as the yield velocity. Provided V < Vy the
interaction is assumed to be elastic. Inserting the above expression for P(z) we get
Er =
8 E∗ z5f
,
15 a∗2
(5.6)
We can relate the critical compression distance zf to the contact yield stress σf of the material
by
σf ≡ p0 (zf ),
From equation (5.2) and (5.3) we have
σf =
2E∗ zf
.
πa∗
(5.7)
Rearranging for zf , and substituting this expression into (5.5) we obtain the energy at which
the particle is estimated to fracture:
Er =
π5 a∗3 σ5f
.
60 E∗4
(5.8)
We shall return to this expression in Section 6, where we shall consider the statistics of particle
collisions leading to fragmentation.
98
5.2
Particle structure
Hertz theory, as mentioned in the previous section, assumes that the two impacting spheres are
solid. However, the particles that make up infant formula, see Figure 1 are clearly hollow. The
study group was asked to investigate how far into the particle there are significant increases in
stress, and therefore whether modelling the spheres as hollow should be incorporated into the
model. Here, we consider the simpler case of two impacting cylinders, although it should be
straightforward to extend the result to two spheres.
Recall that z is the distance from the contact point. If we take the origin of the z-axis to be
at the point where the two cylinders first contact, then z = 0 is the location of a plane of
contact between the two cylinders i.e., the point at which the contact force is distributed over
a rectangle of width 2ac and length equal to the length of the shortest cylinder. This is the
smallest possible area over which the contact force is distributed, and therefore the point at
which the stress has its maximum value.
As the compression increases, the rectangular area becomes larger, because the width of the
rectangle constantly increases due to the curvature of the circular cross-section. Since stress is
force/area, the stresses decrease as z increases.
From [7] we have an expression for σz , the stress in the z direction
ac
σz
,
= −p
p0
a2c + z2
(5.9)
∗ 1
2
where p0 is the maximum pressure, and for two cylinders in contact is given by p0 = PE
πR
(see [7]). (Note that the convention in mechanical engineering is to take compressive stresses
to be negative.)
In Figure 6 we plot a graph of the stress divided by the maximum pressure ( pσ0z ) against the
distance from the contact point, zac , for a constant ac and typical particle parameters.
At z = 0, the compressive stress is at its maximum. We also observe that the stresses diminish
extremely rapidly as you move further away from the contact point. We expect an entirely
analogous result in the case of two impacting spheres. If the stress changes are non negligible
at a depth that is close to the wall thickness, then Hertz theory is not applicable. If the extent
of the zone where there is a measurable increase in stress is small, then Hertz theory can be
used to represent the contact mechanics of these particles. As the walls of the infant formula
particles are approximately a fifth of the diameter, we conclude that approximating the particles
as solid spheres is a reasonable assumption.
99
z 300
ac
250
200
150
100
50
0
−1
−0.8
−0.6
−0.4
σz
p0
−0.2
0
Figure 6: We consider two cylinders in contact. The graph shows the stress divided by the
maximum pressure ( pσ0z ) plotted against the distance from the contact point, ( azc ), for a constant
ac .
6
The statistics of collisions leading to fragmentation
Consider a particle size distribution such that the proportion of particles with radius lying in
the range (a, a + da) is n(a)da. Furthermore take the probability of breaking a particle of
size a when it undergoes a collision in which an energy in the range (E, E + dE) is released to
it is pb (E, a)dE. We note that at present both these distributions will have to be derived from
experiment, though an ultimate goal of this type of work might be to formulate a Smoluchowski
model of the evolution of the size distribution n(a).
In §4 we calculated the transverse velocity ξ̇, due to centrifugal force, of a particle being
advected in a curved pipe as a function of its radius. Notably this was proportional to the
square of the radius of the particle (bigger particles travel more quickly than smaller ones). We
also saw that this velocity slowed markedly as the particle approached the outer edge of the
pipe, a consequence of the slowing of the flow along the pipe in the immediate vicinity of the
pipe wall. This lead us to conclude that collisions of particles with the pipe wall give rise to
insignificant particle fragmentation and to hypothesise that the main cause of particle breakage
is collisions between particles of different radii travelling with different transverse velocities ξ̇.
It is the aim of this section to quantify this process.
We start by considering the total number of collisions between particles of sizes in the range
(a, a + da) with those in the range (α, α + dα). In the direction of the principle normal n they
approach each other with velocity |ξ̇(a)− ξ̇(α)|, see Figure 7. Since they will only collide if their
100
a
|ξ̇(a) − ξ̇(α)|
b
α
Figure 7: A schematic diagram of a particle with radius a approaching a particle with radius
α. b denotes the distance between their centres, and |ξ̇(a) − ξ̇(α)| their approach velocity.
centres come within a distance (a + α) of each other the average number of collisions occurring
to a particle of radius a with particles with radii in the range (α, α + dα) in a small time dt
is Nπ(a + α)2 |ξ̇(a) − ξ̇(α)|n(α)dαdt, where N is the total number density of all particles. It
follows that


The number of collisions in time dt per unit
 volume between particles with radii in range (a, a + da) 
with those with radii in range (α, α + dα)
= N2 π(a + α)2 |ξ̇(a) − ξ̇(α)|n(α)n(a)dαdadt
Of these collisions we need to work out how many lead to fracture. We have already calculated
the energy released in an impact between a particle radius a and one with radius α when their
centres are offset by a distance b (see Figure 7). Such collisions release energy
1
b2
m(a)m(α)
2
Er (a, α, b) =
(v(a) − v(α)) 1 −
.
2 m(a) + m(α)
(a + α)2
Of this we hypothesise that the energy divides between the particles proportional to their mass,
that is
Energy available for
fracture of particle a
Energy available for
fracture of particle α
m(a)
Er (a, α, b),
m(a) + m(α)
m(α)
Er (a, α, b),
=
m(a) + m(α)
=
(6.1)
(6.2)
where Er is the energy at which the particle is estimated to fracture, and is given in Section 5.
The fraction of collisions for which the offset distance between the centres of the particles (at
collision) lies in (b, b + db) is (2πbdb)/(π(a + α)2 ) (see Figure 8b) and the energy released in
such a collision is Er (a, α, b)db.
101
α
b+db b
α
a
a+ α
Figure 8: Here the first figure shows a large particle of size a approaching a smaller particle
size α. The second figure shows the end view of the cylinder with the smaller particle lying at
the centre. Provided the centre of the larger particle (with radius a) lies within the cylinder
the particles will collide.
Referring to (6.2) we find that
Z
Number of particles size
2
range (a, a + da) per unit time = N π (a + α)2 |ξ̇(a)
per unit volume which fracture
Z Emax m(a)
m(a)
pb
−ξ̇(α)|n(a)n(α)
Er d
Er dα. (6.3)
m(a) + m(α)
m(a) + m(α)
0
Translating (6.3) into terms of the separation b between the particle centres (perpendicular to
the velocity) upon impact this can be written as
Number of particles size
Fracture rate = range (a, a + da) per unit time = F(a, x)
per unit volume
Z∞
m(a)(a + α)2
2
|ξ̇(a, x)
= Nπ
0 m(a) + m(α)
Z a+α m(a)
∂Er
−ξ̇(α, x)|n(a)n(α)
pb
Er (a, α, b)
db dα. (6.4)
m(a) + m(α)
∂b
0
7
Conclusions
We have provided a model for the flow in the curved pipe which we used to show that the
velocity of particles decreases rapidly as they approach the pipe wall. We believe that particle102
wall collisions give rise to insignificant particle fragmentation, and that the main mechanism
for particle breakage is collisions between particles of different sizes.
We have analysed inter-particle collisions, using Hertzian contact laws to provide a criterion for
when a binary collision leads to breakup. In addition we have justified modelling these particles
as solid, rather than hollow. Finally, we used a statistical mechanics approach to generate an
expression for the total rate of particle breakup in the flow.
Future work would involve experimental verification of this model.
Acknowledgements
This work was carried out at the 70th European Study Group with Industry, hosted by the
Mathematics Applications Consortium for Science and Industry (MACSI), funded by the Science Foundation Ireland (SFI) Mathematics initiative 06/MI/005. We would like to thank
William Lee for his comments on an earlier draft.
103
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